Random walk models and probabilistic techniques for inhomogeneous polymer chains

Modeling of polymer chains, that is long linear molecules made up of a sequence of simpler units called monomers, has, for a lot of time, received a lot of attention in physics, chemistry, biology, ... Mathematics belongs to this list too. For example, probabilistic models that naturally arise in statistical mechanics have been widely studied by mathematicians for the very challenging and novel problems that they pose. This is true to the extent that, in probability, the word polymer has become synonymous with self–avoiding walk, a basic and extremely difficult mathematical entity. The interaction of a polymer with the environment leads to even more challenging questions: these are often tackled in the framework of directed walks. Restricting attention to directed trajectories is a way of enforcing the self–avoiding constraint that leads to much more tractable models. Still, the interaction with the environment may quickly lead to extremely difficult questions. A particularly interesting situation is that of an inhomogeneous polymer (or copolymer) in the proximity of an interface between two selective solvents. The polymer is inhomogeneous in that its monomers may differ in some characteristics and, consequently, the interaction with the solvents and the interface may vary from monomer to monomer. In interesting cases there can be a phase transition between a state in which the polymer sticks very close to the interface (localized regime) and a state in which it wanders away from it (delocalized regime). The typical mechanism underlying such phase transitions is an energy/entropy competition. The main task of this Ph.D. thesis is to introduce and study random walk models of polymer chains with the purpose of understanding this competition in a deep and quantitative way. Since a random walk can be regarded as an example of an abstract polymer, the idea of modeling real polymers using random walks is quite natural and it has proved to be very successful. The models we are going to consider are modifications of a basic model intro- duced in the late eighties by T. Garel, D. A. Huse, S. Leibler and H. Orland that in turn had translated into the language of theoretical physics ideas that were developing in the applied sciences. Despite the fact that the definition of these models is extremely elementary, their analysis is not simple at all. For a number of interesting issues there is still no agreement in the physical literature. From a mathematical viewpoint it has taken quite a lot of time and effort to rigorously derive their basic properties, and several interesting questions are still open. In this Ph.D. thesis we present new results that answer some of these questions. The approach taken here is essentially probabilistic, and it is interesting to note how the analysis performed has required the application of a wide range of techniques in- cluding Large Deviations and Concentration Inequalities (Ch. 2), Perron–Frobenius Theory (Ch. 3), Renewal Theory (Ch. 4) and Fluctuation Theory for random walks (Ch. 5 and 6). A numerical and statistical study has also been performed (Ch. 2). Reciprocally, the study of the models stimulates the extension of these techniques, see, for instance, the Local Limit Theorem for random walks conditioned to stay positive presented in Chapter 6. The thesis is organized as follows. The definition of the models we consider is given in detail in Chapter 1, where we also give some motivation and we collect the known results from the literature. The following five chapters contain original results. A detailed outline of the thesis may be found in Section 5 of Chapter 1.